3. Let f : R" →R be differentiable and fix ro E R". Assume Vf(ro) + 0, whereVf(ro) is the gradient vector of f at ro. Show that for any v e R" with |v| = 1,we have-|Vf(ro)| < D, f(xo) <|Vf(xo)];with equality if and only ifVf(ro)|V{(ro)|v = +

Answered: 3. Let f : R" →R be differentiable and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Find the gradient of the function f(x, y) = cos(x² + y²) at the point (3,-4). 5. Find the directional derivative of f(x, y) = x³y³ at the point (4,3) in the direction of v= ✓(i+j)
5) P = (−2, − 1) and draw the gradient vector at P. Draw to scale. Sketch the level curve of f(x, y) = x² - y² that passes through :
Would I take the gradient of each A and phi, then find them at the point given before dotting them?? Thank you!!
6. Denote by V the vector ·(). Then we can define the operations grad (f) = f (gradient of f), div (F) = V F (divergence of F), and curl (F) = ▼ × F (curl of F). (a) Given F = (2x + 3y, 3x + z, sin x), calculate div(F) and curl(F). (b) Is the vector field F = (x + y,3y, z-x+1) conservative? What about F = (yz, xz, xy)? (c) Calculate V × (Vf) =curl(grad f) and V. (V × F) = div(curl F).
c) Starting from the point on the surface of the function (1, 1, f(1,1)), use the gradient to determine the maximum rate of change of f and the direction of that maximum change.
2. For the function z = f(x, y) is known f (2, – 1) = -2, f-(2, -1)=-1, f,(2, –1) = 2. Find the following (a) Gradient of f at the point (2, –1) :grad f(2, -1) (b) Directional derivative of f at the point (2, – 1) in the direction of the vector –27+37: fa(2, –1 ). (c) Estimate f(1.99, –1.02) = (d) Differential of f at the point (2, -1)
16. If n is a unit vector, then în · Vf is called the directional derivative of f in the direction n. Find the directional derivative of ƒ (x,y) = x²yz in the direction 4 î - 3 k at a point (1,-1, 1).
Let a be a real number and f(r, y) = 3ar'y + 5zy. Let v = 3i – j and let Vf(z, y) denote the gradient of f(r, y). i) Find Vf(1, 1) in terms of a. in) If the rate of change of f(z, y) at the point (1, 1) in the direction of v is- V2 -6/5 find a. Türkçe: a bir reel sayı olmak üzere f(z, y) = 3az?²y + 5æy³ şeklinde tanımlansın. %3D v = 3i – j ve Vf(x, y). f(x, y) fonksiyonunun gradyanı olsun. i) Vf(1,1) değerini a değişkeninine göre ifade ediniz. i) f(r, y) fonksiyonunun (1, 1) noktasında v yönünde değişim oranın hızı -6/5 ise a değerini bulunuz. O i) (6a + 5) i + (3a + 15) j, i) -2 O i) (6a + 5) i + (3a + 15) j, i) -6 O i) (6a + 5) i + (3a + 15) j, i) 6 O i) (6a + 5) i + (3a + 15) j, ii) 0 i) (3a + 5) i + (-6a + 15) j, i) -2 O ) (3a + 5) i + (-6a + 15) j, i) -6 O ) (3a + 5) i + (-6a + 15) j, i) 6 O i) (3a + 5) i + (-6a + 15) j, i) 0
3. (b) Let f(x, y) = log((ry - 2)²), u =(³, 4). i. Calculate the directional derivative Vaf (3, 1). ii. Determine the unit vector (direction) u for which Vaf (3, 1) is maximal.
Let 2, if (x, Y, z) # (0,0, 0) if (x, y, z) = (0,0, 0) xyz x²+y²+z² > f(x, y, z) = 0, (a) Use the definition to find the directional derivative of the function f at (0,0, 0) in the direction (a, b, c), where a² + 62 + c² = 1. (b) Use Lagrange Multipliers to find the maximal and minimal directional derivatives.

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3. Let f : R" →R be differentiable and fix ro € R". Assume Vf(ro) 0, where Vf(ro) is the gradient vector of f at ro. Show that for any v e R" with Ju = 1, we have -|Vf(ro)| < D.f(ro) < |Vf(r0)l; with equality if and only if Vf(ro) v =+ |Vf(ro)|